Lecture II: Inertia, Diffusion and Dynamics of a Driven Skyrmion Schütte, Isagawa, A.R., Nagaosa (PRB 2014) Changing topology: Emergent magnetic Monopoles Milde, Köhler, Seidel,Eng, Bauer, Chacon, Pfleiderer, Buhrandt, A. R., Science (2013) skyrmions in chiral magnets , Tallahassee 1/14 reminder I: skyrmion phase skyrmions in chiral magnets , Tallahassee 1/14 reminder II: emergent electrodynamics & topological quantization 1 • effective electric charge: e q / = spin parallel/antiparallel to local magnetization ↓ ↑ ∓2 • emergent magnetic & electric fields: e ~ Berry phase for loops Bi = ²ijknˆ (∂jnˆ ∂knˆ) in space 2 · × Berry phase for loops e E = ~ nˆ (∂inˆ ∂tnˆ) in space-time i · × • topological quantization: winding number -1 one flux quantum per skyrmion skyrmions in chiral magnets , Tallahassee 1/14 Emergent electric and magnetic fields directly experimentally measurable But: no conventional photons in skyrmion phase why: charged matter (skyrmions) hang around charged Wigner crystal in magnetic fields: quadratic dispersion! skyrmions in chiral magnets , Tallahassee 1/14 Skyrmion lattices and single skyrmions Tokura, Nagaosa et al. 2010 Wiesendanger group, Science 2013 writing of single skyrmions now: single skyrmion in two dimensions in ferromagnetic background skyrmions in chiral magnets , Tallahassee 1/14 classical (or quantum) point particle in a medium needed: mass, friction, effective magnetic field, coupling to external fields, internal excitations,…. skyrmions in chiral magnets , Tallahassee 1/14 first try: guess Newton’s equation by symmetry for slow motions (low fequencies) first time derivative second time external forces from derivative currents, field gradients, thermal fluctuations skyrmions in chiral magnets , Tallahassee 1/14 first try: guess Newton’s equation by symmetry for slow motions (low fequencies) gyrocoupling friction inertia new: = “gyrodamping” effective magnetic field = Magnus force skyrmions in chiral magnets , Tallahassee 1/14 Landau-Lifshitz-Gilbert Gleichung semiclassical dynamics of magnetization without current, precession in damping due to spin-orbit effective fields (static + thermal noise) how modified by current? What terms allowed by symmetry? skyrmions in chiral magnets , Tallahassee 1/14 Landau-Lifshitz-Gilbert equation velocity of spin-currents Berry phase precession in damping due to spin-orbit spin torques effective fields Berry phase action 3 describes both SB = d rdt MA~(ˆn)(∂t + ~vs∇~ )ˆn topological Hall effect Z and main forces on magn. structure Monopole vector field counts area on surface skyrmions in chiral magnets , Tallahassee 1/14 Not covered: extra damping terms e spin velocity: vs = j /M (Zhang, Li, 2004) emergent precession in damping due to spin-orbit electromagnetism, effective fields Magnus forces ohmic damping due to currents induced by emerging electric field (Zhang,Zhang 09, Zang et al. 11) higher gradients but: no spin orbit needed, enhanced by factor kF l skyrmions in chiral magnets , Tallahassee 1/14 first try: guess Newton’s equation by symmetry for slow motions (low fequencies) first: consider only linear time derivative (Thiele 1973) skyrmions in chiral magnets , Tallahassee 1/14 Effective equation for skyrmion coordinate (Thiele equation) . ansatz: static skyrmion at position R(t): . to project equation on translational motion: multiply with and integrate over space: = spin density * winding number skyrmions in chiral magnets , Tallahassee 1/14 strongest force: Berry-phase coupling to electric currents, = Magnus force = gyrocoupling (Thiele) “gyrocoupling“ skyrmion electronic drift-velocity velocity ( spin-current /magnetization ) skyrmions in chiral magnets , Tallahassee 1/14 first try: guess Newton’s equation by symmetry for slow motions (low fequencies) now: effective mass + frequency dependent damping problem: frequency dependencies strong violates causality (wrong sign = antidamping) therefore: full frequency dependence needed! skyrmions in chiral magnets , Tallahassee 1/14 frequency dependent dynamics of skyrmions in d=2 (linear response) velocity velocity of of field thermal skyrmion spin-currents gradients fluctuations strongly frequency-dependend screening of external forces damping gyrocoupling effective mass skyrmions in chiral magnets , Tallahassee 1/14 frequency dependent dynamics of skyrmions in d=2 (linear response) velocity velocity of of field thermal skyrmion spin-currents gradients fluctuations key for identification of dynamics: fluctuation-dissipation theorem skyrmions in chiral magnets , Tallahassee 1/14 Simulations (classically): Landau Lifshitz Gilbert equation including thermal fluctuations with , , Garcia-Palacios, Lazaro (1998) skyrmions in chiral magnets , Tallahassee 1/14 simulations in classical limit linear equation: where is, e.g., mass coming from? excitation of moving skyrmion stores “kinetic” energy leads to retardation effects skyrmions in chiral magnets , Tallahassee 1/14 effective mass friction spin-wave excitations gyrodamping Gyrocoupling = effective magn. field units: gap of ferromagnet (~GHz) skyrmions in chiral magnets , Tallahassee 1/14 effective mass • strong frequency dependencies set by excitations of spin wave • huge effectiv mass (~ number of flipped spins of skyrmion) Bad news? No fast skyrmion dynamics? No! Depends on how skyrmion is manipulated! skyrmions in chiral magnets , Tallahassee 1/14 thermal fluctuations or skyrmion flows approximately with external forces by field gradients electric current, only weak excitation excite internal modes of external modes large mass, delayed response small mass, fast response skyrmions in chiral magnets , Tallahassee 1/14 “apparent mass” depends on driving mechanism controlled by frequency dependence of screening apparent dynamics matrix skyrmions in chiral magnets , Tallahassee 1/14 apparent mass: diffusive motion driven by field gradient current driven: tiny mass skyrmions in chiral magnets , Tallahassee 1/14 “nice to have” properties of skyrmions • small “apparent” mass for ultra-fast manipulations by currents • small friction • tiny thermal diffusion constant (precession in huge effective B-field) skyrmions in chiral magnets , Tallahassee 1/14 Quantum dynamics of skyrmions Garst, Schütte, 2014 • only relevant in insulators (e.g. Cu2OSeO3) at T<< gap • Skyrmion: dynamics in effective B-field spin-wave skyrmion lives in single flat Landau level continuum • close to boundary: chiral edge states coherent quantum dynamics close to edge state • also interesting: skyrmion & topological insulators, topological superconductors,… skyrmions in chiral magnets , Tallahassee 1/14 conclusions: skyrmions as particles . large intrinsic effective mass of skyrmions . dynamics strongly frequency dependent due to emission of spin-waves . nevertheless: rapid manipulation by currents possible . quantum dynamics: particle in Landau level, edge states . skyrmions & obstacles: pinning & depinning controllable by magnetic fields and currents speeding up by defects Schütte, Isagawa, Rosch, Nagaosa (2014) Jan Müller, Achim Rosch (2014) skyrmions in chiral magnets , Tallahassee 1/14 Back to 3d: destroying skyrmions & changing topologogy emergent magnetic monopoles Milde, Köhler, Seidel,Eng, Bauer, Chacon, Pfleiderer, Buhrandt, A. R., Science (2013) skyrmions in chiral magnets , Tallahassee 1/14 destruction of the skyrmion phase experiment: track by magnetic force microscopy skyrmions on surface of Fe0.5Co0.5Si Milde, Köhler, Seidel, Eng, TU Dresden step 1: cool system down to 10 K at B=20 mT measure z-component of magnetization by magnetic force microscopy ∆f skyrmions in chiral magnets , Tallahassee 1/14 destruction of the skyrmion phase experiment: track by magnetic force microscopy skyrmions on surface of Fe0.5Co0.5Si Milde, Köhler, Seidel, Eng, TU Dresden step 1: cool system down to 10 K at B=20 mT measure z-component of magnetization by magnetic force microscopy result: metastable skyrmion lattice, slightly disordered good contrast due to low temperature few fluctuations (high topolog. stability) skyrmions in chiral magnets , Tallahassee 1/14 destruction of the skyrmion phase step 2: destroy skyrmion lattice by reducing B-field observation: neighboring skyrmions merge, forming elongated objects skyrmions in chiral magnets , Tallahassee 1/14 further reducing B: longer and longer linear structures form by combining skyrmions realizing helical state with large number of defects local helix skyrmions in chiral magnets , Tallahassee 1/14 What happens in the bulk? How does topology change ? skyrmions in chiral magnets , Tallahassee 1/14 Surface reflects bulk behavior surface: FT of MFM bulk: neutrons numerical simulations: neutron scattering skyrmions in chiral magnets , Tallahassee 1/14 emergent electrodynamics: winding number of skyrmions = one flux quantum of emergent magnetic field needed to change winding number: sources and sinks of emergent magnetic field = quantized magnetic charges = emergent magnetic monopoles and antimonopoles skyrmions in chiral magnets , Tallahassee 1/14 Historical remarks on magnetic monopoles Paul Dirac (1931): why is charge quantized? magnetic charge = magnetic monopole would enforces charge quantization „Dirac string“ invisible only if both electric and magnetic charge are quantized 2π~ 2π~ qm = n e = n e qm skyrmions in chiral magnets , Tallahassee 1/14 Historical remarks on magnetic monopoles Paul Dirac (1931): why is charge quantized? magnetic charge = magnetic monopole would enforces charge quantization t‘Hooft (1974), Polyakov (1974): magnetic